Nontrivial phase matching in helielectric polarization helices: Universal phase matching theory, validation, and electric switching

Significance Nonlinear three-wave mixing is a fundamental physical pathway in nonlinear optical materials, providing a route for producing a high-energy photon through two input photons. This report proposes a generalized phase-matching theory that evaluates second-order nonlinear optical properties for arbitrary polarization structures. It predicts light amplification pathways by using polarization helices, which are justified by experiments based on spontaneous helielectric nematic fluids. The unique polar electric field switching of the helielectrics enables the dynamic tuning of the emitted nonlinear light. The discovery not only introduces a general principle for nonlinear optical calculation but also contributes to a significant advance in diversifying the category and structure of polar materials.


Supplementary Discussion 1. Transfer matrix calculation
To calculate the propagation of light, we employed the 4×4 Mueller matrix formalism. The procedure for evaluating the light traveling through the medium is described in the following. We first consider the refractive index tensor in the medium as and are the ordinary and the extraordinary refractive index, respectively. In our optical simulation, director fields are produced by functions for the poling (PP) and helielectric (HN*) structures. By using this structural information, the dielectric permittivity tensor is generated for each director orientation as follows. The rotation matrix is created from the rotation angle of the director from the reference axis (β), so that the effective dielectric permittivity tensor becomes: where ( ) is the rotation matrix that rotates by angle of β. The elements of Maxwell's equations in Cartesian coordinates are written as follows: where is the speed of light, , the components of electric and magnetic field and , the electric and magnetic flux density, respectively. When the 6×6 matrix, the matrix of the electric and magnetic field vector and the matrix of the electric and magnetic flux density are abbreviated as , and , Eq. (1) is described as Moreover, a linear relationship can be expressed between and via the matrix in which the electric permittivity and the magnetic permeability information are included: = .
(3) By defining the time displacements of and as exp ( ), the following spatial wave equation can be obtained from Eqs. (2,3), where is the angular frequency of light and the spatial component of (supposing is composed of the spatial and time components). Considering that the light incidents on the xz plane as the plane of incidence, then the components of the are modified as follows: The x component of the wavenumber vector for the incident light is = sin , where is the refractive index of incident side medium and the angle between normal direction of medium surface and the direction of the light propagation. Finally, by eliminating components Γ 3 , Γ 6 using Eqs. (4,5), the differential equation is written in the form of the 4×4 matrix, so-called the Berreman's equation (Ref. (1) where are the tangential components of the electric and magnetic fields projected to the incident plane of electromagnetic waves. When light propagates along z axis, is expressed as a 4×4 matrix： in is defined as = = sin . The general solution of the Berreman's equation is given by where D is the propagation length in the medium. Introducing two matrices and to satisfy the following formula, We also introduce the partial transfer matrix: (− ) = exp ( (− )).
By using these quantities, the relationship linking the involved electric field is: The electric field of incident light is represented by and , the electric field of transmitted light by and , and the electric field of reflected light as and .
According to Eqs. (7-9), the following relationships are derived: where is the angle between the normal direction of medium surface and the propagation direction of the transmitted light, and the refractive index of the transmitted-side medium. Therefore, and −1 matrices are given as: Consider light propagation in a single uniform optical layer, the transfer matrix, is defined as follows.
≡ −1 (− ) . Then, for multi-layer structure, is calculated as: where is the total number of the layers and the index of the layers. Using , the relationship between incident, reflected and transmitted light reads: The amplitude ratio of the transmitted to incident light for each polarization condition can be expressed by the following relationship. By specifying the incident light condition, the polarization and phase condition of the light at each position in the medium can be determined. As a result, the propagation of fundamental light and second harmonic light is calculated.

Supplementary Discussion 2. The doping effect of chiral generators on SH signal
In our previous experiment, we found that the SH signal decreases very fast with doping the chiral dopant (Ref.). Two potential effects might coexist here. The first is the diminish of the real polarity of the materials upon the doping of less-polar chiral dopants. The second is the reduction of the effective polarity probed by SHG. Second possibility should be mainly attributed to the decrease of the helical pitch. When the pitch is approaching to wavelength of the input fundamental light, the SH signal should decrease because light "sees" a cancelled polarization field. To test this, we have conducted the SH signal as a function of concentration of chiral dopants by using chiral dopants with distinct chiral strength. Here we compare chiral dopants R5011 and S1. R5011 have much stronger helical twisting power than S1. Figure S5 shows that the concentration dependences of SH signal in both RM734/R5011 and RM734/S1 systems (data of RM734/S1 reproduced from Ref. (2)). As seen, when doping of R5011 is less than 1 wt%, the SH signal almost vanishes, where the polarity of the system is almost same to that of RM734 because the doping ratio is very small. On the other hand, S1-doped system show a much slower decrease of SH signal. When plotting the helical pitch dependences of the SH signal, the results from the two systems are well-consistent: the SH signal shows almost same helical pitch dependence. It means the observed SH signal reduction should mainly come from the second effect, i.e., the reduction of the effective polarity probed by SHG reduces. When the doping is small, the system polarity remains almost unchanged with the neat ferroelectric nematic phase. Especially, a clear SHG enhancement was also observed in RM734/R5011 system when the helical pitch is close to the SH wavelength of 532nm. This is also consistent with our previous results in RM734/S1 systems. Fig. S1. Coherence length measurement of RM734 at 110 °C in a syn-parallel rubbed wedge cell. The incident polarization of the fundamental beam is parallel to the polarization. The coherence length and are obtained by fitting.  Only a sine-function of the phase factor of is considered. In (A) and inset, the dashed red, green dashed and blue dashed lines correspond to the traditional quasi-matching condition for the periodic poled structure with m=1, 2, 3, respectively. The red solid line corresponds to the present quasi-matching condition for the helielectric structure with m=1. The blue solid and green solid lines correspond to the helielectric structures with m=2, 3 (Inset). In (B), the expanded views of specific area of the curves are shown in the corresponding insets. Such a situation in HN* state never happen in the experiment.     S7. The concentration (A) and helical pitch (B) dependencies of SH signal in both RM734/R5011 and RM734/S1 systems (data of RM734/S1 reproduced from Ref. (2)). R5011 is a commercial chiral dopant. The SH efficiency is defined a s the intensity ratio of sample SH signal to the SH signal of a reference Y-cut quartz.